58 Prof. Challis on the "Loss of half an undulation" 



that position a decrement of density from within to without, due 

 partly to the original waves, and partly to the lateral spreading. 

 So far as the decrement is due to the original waves, the propa- 

 gation from within to without tends continually to increase the 

 density at the mouth of the tube. But this tendency is imme- 

 diately counteracted by the lateral spreading, the effect of which, 

 consequently, in producing decrement of density from within to 

 without near the tubers mouth, has at each instant a given ratio 

 to the decrement due to the original waves. Analogous consi- 

 derations apply to the case in which the particle is urged by an 

 accelerative force from without to within. In this case the den- 

 sity increases from within to without, and, so far as the increment 

 is due to the original waves, the tendency of propagation is to 

 diminish the density at the mouth. This diminution is imme- 

 diately counteracted by the external lateral spreading, and the 

 increment of density from within to without, due at any instant 

 to the original waves, is thus augmented, and in the same con- 

 stant ratio as in the former case. Hence, if the accelerative force 

 impressed on the particle by the original waves be /, the actual 

 accelerative force is qf, q being a certain constant. In order 

 that the density at the mouth of the tube may be always equal 

 to that of the external air, or that the position of what is called 

 a loop in this class of vibrations be at the mouth, it is necessary 

 that the factor q be equal to 2. When this is the case, the 

 motion within the tube, as the mathematical theory proves, con- 

 sists of two equal series of undulations propagated in opposite 

 directions. Consequently, as the motion is excited at one end 

 of the tube, the effect that takes place at the opposite open end 

 is a reflexion of the undulations in a phase contrary to that of 

 incidence. A reflexion of the same kind would be produced by 

 placing a diaphragm transverse to the axis of a tube along which 

 a series of waves is propagated, and giving to it just double the 

 velocity which the fluid in contact with it would receive from 

 the undisturbed effect of the propagated waves. The diaphragm 

 will thus impress on the fluid an accelerative force equal to that 

 due to the waves, and give rise to two additional series of waves, 

 equal in magnitude to the original series ; one of opposite phase 

 and propagated in the opposite direction, and the other of the 

 same phase and propagated in the same direction. By giving 

 to the diaphragm any other motion having a given ratio, q, to the 

 undisturbed motion, these effects may be altered at pleasure. 

 If (/ = 1, no effect is produced, and if q be less than unity, the 

 return waves are of the same phase as the incident. 



In order to apply these considerations to the reflexion of 

 light at the surfaces of transparent media, I shall suppose, for 

 the sake of simplicity, that the light is incident perpendicularly 



